MATHEMATICS

V. A. IL’IN

THE FOURIER METHOD FOR A HYPERBOLIC EQUATION WITH DISCONTINUOUS COEFFICIENTS

(Presented by Academician I. G. Petrovskii on 30 VI 1961)

In the present paper, by means of the Fourier method we prove the existence of a classical solution of a mixed problem for a hyperbolic equation with discontinuous coefficients. Here the surfaces on which the coefficients of the equations are discontinuous are subjected only to the Lyapunov condition, while the coefficients of the equations, the boundary surface, and the data of the problem are subjected to such smoothness requirements which, in the particular case when discontinuities of the coefficients are absent, pass into the conditions found in the work \((^1)\), i.e. into the sharpest of the conditions known up to now for the solvability of the mixed problem for a hyperbolic equation with smooth coefficients.

1°. Let an \(N\)-dimensional open domain \(g\), together with its boundary \(\Gamma\), be contained in some open domain \(T\). Suppose further that inside the domain \(g\) there lie \(k\) closed pairwise nonintersecting surfaces \(C_1, C_2, \ldots, C_k\), dividing the domain \(g\) into \(k+1\) domains \(g_0, g_1, \ldots, g_k\). (We denote by \(g_i\) \((i=1,2,\ldots,k)\) the domain lying inside the surface \(C_i\).) Consider \(k+1\) self-adjoint elliptic operators

\[ L_l u=\sum_{i,j=1}^{N}\frac{\partial}{\partial x_i}\left[a_{ij}^{(l)}(x)\frac{\partial u}{\partial x_j}\right]+c^{(l)}(x)u, \tag{1} \]

defined in the following domains: \(L_0\) in \((T-g_1-\cdots-g_k)\), \(L_i\) \((i=1,2,\ldots,k)\) in \((g_i+C_i)\).

Let us denote by the symbol \(\Omega\) the open cylinder \(\Omega=g\times(0<t<t_0]\), and by \(\Omega_l\) \((l=0,1,\ldots,k)\) the open cylinder \(\Omega_l=g_l\times[0<t<t_0]\); by the symbols \(\overline{\Omega}\) and \(\overline{\Omega}_l\) the corresponding closed cylinders; by the symbol \(S_i\) \((i=1,2,\ldots,k)\) the cylindrical surface \(S_i=C_i\times[0\le t\le t_0]\); and by the symbol \(S_0\) the cylindrical surface \(S_0=\Gamma\times[0\le t\le t_0]\).

Consider in the cylinder \(\Omega\) the mixed problem for a hyperbolic equation with discontinuous coefficients:

\[ L_l u-u_{tt}=-f(x,t)\quad \text{in } \Omega_l\ (l=0,1,\ldots,k), \]

\[ u(x,0)=\varphi(x),\qquad u_t(x,0)=\psi(x), \tag{2} \]

\[ u\big|_{S_0}=0,\qquad [u]\big|_{S_i}=0,\qquad \left[\frac{\partial u}{\partial \nu}\right]\Bigg|_{S_i}=0 \quad (i=1,2,\ldots,k); \]

where

\[ [u]\big|_{S_i}=u\big|_{S_i-0}-u\big|_{S_i+0},\qquad \left[\frac{\partial u}{\partial \nu}\right]\Bigg|_{S_i} = \frac{\partial u}{\partial \nu_i}\Bigg|_{S_i-0} - \frac{\partial u}{\partial \nu_0}\Bigg|_{S_i+0}; \]

the symbols \(S_i-0\) and \(S_i+0\) mean that the limiting values are taken respectively from the inner and from the outer (with respect to the cylinder \(\Omega_i\)) sides of the surface \(S_i\); \(\nu_i\) denotes the conormal, inner with respect to the domain \(\Omega_i\), for the operator \(L_i\) \((i=1,2,\ldots,k)\); \(\nu_0\) denotes the conormal, outer with respect to \(\Omega_0\), for the operator \(L_0\).

It is natural to call a classical solution of problem (2) a function \(u(x,t)\) satisfying the following requirements: 1) \(u(x,t)\) is continuous in the closed cylinder \(\overline{\Omega}\); 2) all first and second derivatives ...

derivatives of the function \(u(x,t)\) are continuous in each of the open cylinders \(\Omega_l\) \((l=0,1,\ldots,k)\); 3) the first-order derivatives are continuous in \((\Omega_i+S_i)\) \((i=1,2,\ldots,k)\) and in \((\overline{\Omega}_0-S_0)\); 4) the derivative \(u_t(x,t)\) is continuous for \(t=0,\ x\in g\); 5) \(u(x,t)\), in the classical sense, satisfies all the conditions of problem (1).

In [2] we established conditions A under which there exists a complete orthonormal system of classical eigenfunctions of the problem with discontinuous coefficients*

\[ L_l v+\lambda v=0 \quad (\text{in } g_l,\ l=0,1,\ldots,k), \]

\[ v\big|_{\Gamma}=0,\qquad [v]\big|_{C_i}=0,\qquad \left[\frac{\partial v}{\partial \nu}\right]\bigg|_{C_i}=0 \quad (i=1,2,\ldots,k). \tag{3} \]

In the same paper it is proved that, under conditions A, the complete orthonormal system of classical eigenfunctions of problem (3) coincides with the complete system of generalized eigenfunctions of this problem.

In complete analogy with what is set forth in Ch. 2 of [1], the following two theorems are proved.

Theorem 1. If conditions A are fulfilled, then there can exist only one classical solution of the mixed problem (2).

Theorem 2. Let conditions A be fulfilled and, in addition, the following conditions: 1) \(\varphi\in C^{(0)}\) in \((\overline g+\Gamma)\), \(\varphi\in C^{(1,\mu)}\) in \(g_0\); 2) \(\psi\in C^{(0)}\) in \((\overline g+\Gamma)\); 3) \(f\in C^{(0)}\) in \(\overline{\Omega}\). Then, if there exists a classical solution of problem (2), this solution belongs to \(W_2^1(\overline{\Omega})\) and coincides with the generalized, in the sense of O. A. Ladyzhenskaya ([3], pp. 72–73), solution of problem (2).

It is known that the mixed problem (2) describes the vibrations of a bounded volume \(g\) with inhomogeneous filling. From this point of view, Theorem 2 proves the existence, for the vibrating particles, of finite energy for almost all moments of time \(t\) from the segment \(0\le t\le t_0\). We note that, for the proof of Theorem 2, in addition to the method of [1] and the results of [2], Theorem 2 of [4] is also used essentially.

\(2^\circ\). We pass to the proof of existence of a classical solution of the mixed problem (2). If the above conditions A are fulfilled, i.e., if there exist classical eigenfunctions of problem (3), then one may try to solve the mixed problem (2) by the Fourier method. Formal application of the Fourier method leads to the following series:

\[ u(x,t)=\sum_{n=1}^{\infty} v_n(x)\left\{\varphi_n\cos\sqrt{\lambda_n}t +\frac{\psi_n}{\sqrt{\lambda_n}}\sin\sqrt{\lambda_n}t +\right. \]

\[ \left. +\frac{1}{\sqrt{\lambda_n}}\int_0^t f_n(\tau)\sin\sqrt{\lambda_n}(t-\tau)\,d\tau\right\}. \tag{4} \]

Here \(\lambda_n\) are the eigenvalues, \(v_n(x)\) the corresponding orthonormal eigenfunctions of problem (3), and \(\varphi_n,\psi_n\), and \(f_n(t)\) are the Fourier coefficients in the expansion of the functions \(\varphi(x),\psi(x)\), and \(f(x,t)\) with respect to the system \(v_n(x)\).

Definition. Let a certain function \(F(x)\) \((F(x,t))\) belong to the class \(W_2^{(2)}\) in each of the domains \(\overline g_l\) \((\overline{\Omega}_l)\) \((l=0,1,\ldots,k)\). We shall say that this function satisfies the conjugation conditions on the surfaces \(C_i\) \((S_i)\), if almost everywhere on \(C_i\) \((S_i)\) there hold

* Conditions A are as follows: 1) the surface \(\Gamma\) is regular, the surfaces \(C\) belong to the Lyapunov class; 2) the coefficients of the operators \(L_l\) belong to the classes \(a_{ij}^{(l)}\in C^{(1,\mu)}\), \(C^{(l)}\in C^{(0,\mu)}\) in the domains of definition of the operators and, moreover, \(C^{(l)}\le 0\).

equalities

\[ [F(x)]\big|_{C_i}=0,\qquad \left[\frac{\partial F}{\partial \nu}\right]\bigg|_{C_i}=0 \qquad \left( [F(x,t)]\big|_{S_i}=0,\qquad \left[\frac{\partial F}{\partial \nu}\right]\bigg|_{S_i}=0 \right) \]

\[ (i=1,2,\ldots,k). \]

Suppose that the following four conditions are satisfied:

1) The surface \(\Gamma\) is regular\(^*\), and the surfaces \(C_i\) \((i=1,2,\ldots,k)\) satisfy the Lyapunov conditions.

2) The coefficients of the operators \(L_l\) \((l=0,1,\ldots,k)\), first, satisfy the conditions A indicated above and, in addition, in the closed domain \(g_l\) \((l=0,1,\ldots,k)\) the coefficients \(a_{ij}^{(l)}\) have continuous\(^ {**}\) derivatives up to order \([N/2]+2\), and the coefficients \(c^{(l)}\) up to order \([N/2]+1\).

3) \(\varphi\in W_2^{([N/2]+3)}\), \(\psi\in W_2^{([N/2]+2)}\) in each of the domains \(g_l\) \((l=0,1,\ldots,k)\), and, moreover, \(\varphi(x)\) and \(\psi(x)\) are such that each of the functions\(^ {***}\) \(\varphi,L\varphi,\ldots,L^{[(N+4)/4]}\varphi;\psi,L\psi,\ldots,L^{[(N+2)/4]}\psi\) satisfies the conjugacy conditions on the surfaces \(C_i\) and belongs to \(\overset{\circ}{D}(g)\).

4) \(f\in W_2^{([N/2]+2)}\) in each of the cylinders \(\overline{\Omega}_l\) \((l=0,1,\ldots,k)\) and, moreover, \(f(x,t)\) is such that each of the functions \(f,Lf,\ldots,L^{[(N+2)/4]}f\) satisfies the conjugacy conditions on the surfaces \(S_i\) and belongs to \(\overset{\circ}{D}_1(\Omega)\).

We shall call these four conditions conditions B.

Theorem 3. If conditions B are satisfied, then there exists a (and moreover unique) classical solution of the mixed problem (2), defined by the series (4).

In passing, together with the proof of Theorem 3 we give a justification of the Fourier method for problem (2). Namely, we establish that, under conditions B, the series (4) itself, as well as the series obtained by differentiating the series (4) once and twice with respect to \(t\), converge uniformly in the closed cylinder \(\overline{\Omega}\); the series obtained by differentiating (4) once with respect to any of the coordinates \(x_1,x_2,\ldots,x_N\) converge uniformly in each of the cylinders \((\Omega_i+S_i)\) \((i=1,2,\ldots,k)\) and in the cylinder \(\Omega_0'\), where \(\Omega_0'\) denotes the closed cylinder \(\overline{\Omega}_0\) from which an arbitrarily small neighborhood of the surface \(S_0\) has been removed; finally, the series obtained by termwise differentiating the series (4) twice with respect to any variables converge uniformly in any strictly interior subdomain of each of the cylinders \(\Omega_l\) \((l=0,1,\ldots,k)\).

Let us briefly outline the proof of Theorem 3 and the justification of the Fourier method. Invoking the estimate of the Green function of the Dirichlet problem with discontinuous coefficients, obtained in work \((^2)\), and obtaining, by analogous means, estimates for the derivatives of the Green function, we, using the method of Ch. 3 of work \((^1)\), arrive at the following lemma.

Lemma 1. Let conditions A be satisfied. Then the bilinear series of eigenfunctions of problem (3) of the form

\[ \sum_{n=1}^{\infty}\frac{v_n^2(x)}{\lambda_n^{[N/2]+1}} \]

converges uniformly in the closed

\(^*\) The surface \(\Gamma\) is called regular if, in the domain bounded by this surface, the Dirichlet problem for the Laplace equation is solvable for any continuous boundary function.

\(^ {**}\) For a star-shaped domain \(g\), the requirement of continuity of the derivatives of the coefficients \(a_{ij}^{(l)}(x)\), and \(c^{(l)}(x)\) may be replaced by the requirement \(a_{ij}^{(l)}\in W_2^{([N/2]+2)}\), \(c^{(l)}\in W_2^{([N/2]+1)}\) in each of the domains \(\overline{g}_l\) \((l=0,1,\ldots,k)\). We note that the latter requirement is less restrictive than that obtained by S. L. Sobolev \((^5)\), p. 222, for the solvability of the Cauchy problem with smooth coefficients, and than that obtained by O. A. Ladyzhenskaya \((^3)\), p. 101, for the solvability of the mixed problem with smooth coefficients.

\(^ {***}\) Here by the operator \(L\) one should understand the operator with discontinuous coefficients, equal to \(L_l\) in the domain \(g_l\) \((l=0,1,\ldots,k)\).

of that domain \((g+\Gamma)\); the bilinear series of first derivatives of the eigenfunctions of the form

\[ \sum_{n=1}^{\infty} \frac{[\partial v_n(x)/\partial x_i]^2}{\lambda_n^{[N/2]+2}} \]

converges uniformly in each of the domains \((g_i+C_i)\) \((i=1,2,\ldots,k)\) and in the domain \(g_0'\), where \(g_0'\) denotes the closed domain \(\bar g_0\) from which an arbitrarily small neighborhood of the surface \(\Gamma\) has been removed; finally, the bilinear series of second derivatives of the eigenfunctions of the form

\[ \sum_{n=1}^{\infty} \frac{[\partial^2 v_n(x)/\partial x_i\partial x_j]}{\lambda_n^{[N/2]+3}} \]

converges uniformly in any strictly interior subdomain of each of the domains \(g_l\) \((l=0,1,\ldots,k)\).

In complete analogy with what was set forth in Chapter 4 of paper \((^1)\), one proves

Lemma 2. If conditions B are fulfilled, then one may assert the convergence of the numerical series

\[ \sum_{n=1}^{\infty}\varphi_n^2\lambda_n^{[N/2]+3},\qquad \sum_{n=1}^{\infty}\psi_n^2\lambda_n^{[N/2]+2},\qquad \sum_{n=1}^{\infty}\lambda_n^{[N/2]+2}\int_{0}^{t} f_n^2(\tau)\,d\tau \]

and the uniform convergence in the closed cylinder of the series

\[ \sum_{n=1}^{\infty} f_n(t)v_n(x). \]

After this, series (4) and the series obtained by differentiating (4) once and twice are investigated by elementary means analogous to those set forth in Chapter 5 of paper \((^1)\).

Remark 1. The existence of a classical solution of the mixed problem for a hyperbolic equation with discontinuous coefficients of a more general form than (2) was proved in the work of Yu. V. Egorov \((^6)\), but only under the assumption that the coefficients of the equation and the surfaces \(C_i\) of discontinuity of the coefficients satisfy smoothness conditions, extremely high in comparison with conditions B, which grow without bound as the number of dimensions increases.

Remark 2. The results obtained in the present work carry over to the case when part of the surfaces \(C_i\) lies inside others, and also to the case of boundary conditions of the second and third kind

\[ (\partial u/\partial \nu_0 + hu)\big|_{s_0}=0, \]

where \(h(x)\ge 0\) (in this case it is additionally required that \(\Gamma\) satisfy the Lyapunov conditions).

The author expresses gratitude to A. N. Tikhonov for his attention to the work and to I. A. Shishmarev for a number of comments.

Moscow State University
named after M. V. Lomonosov

Received
15 VI 1961

REFERENCES

\(^6\) V. A. Il’in, UMN, 15, issue 2, 97 (1960).
\(^2\) V. A. Il’in, DAN, 137, No. 2 (1961).
\(^3\) O. A. Ladyzhenskaya, The Mixed Problem for a Hyperbolic Equation, 1953.
\(^4\) V. A. Il’in, DAN, 137, No. 1 (1960).
\(^5\) S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics, L., 1950.
\(^6\) Yu. V. Egorov, DAN, 134, No. 3 (1960).